# Joint estimation of parameters in Ising model

We study joint estimation of the inverse temperature and magnetization parameters (β,B) of an Ising model with a non-negative coupling matrix A_n of size n× n, given one sample from the Ising model. We give a general bound on the rate of consistency of the bi-variate pseudolikelihood estimator. Using this, we show that estimation at rate n^-1/2 is always possible if A_n is the adjacency matrix of a bounded degree graph. If A_n is the scaled adjacency matrix of a graph whose average degree goes to +∞, the situation is a bit more delicate. In this case estimation at rate n^-1/2 is still possible if the graph is not regular (in an asymptotic sense). Finally, we show that consistent estimation of both parameters is impossible if the graph is Erdös-Renyi with parameter p>0 free of n, thus confirming that estimation is harder on approximately regular graphs with large degree.

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